Problem: All of the 3rd grade teachers and students from Gardner Bullis went on a field trip to an art museum. Tickets were $$7.00$ each for teachers and $$2.50$ each for students, and the group paid $$53.00$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$21.00$ each for teachers and $$9.50$ each for students, and the group paid $$179.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7x+2.5y = 53}$ ${21x+9.5y = 179}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-21x-7.5y = -159}$ ${21x+9.5y = 179}$ Add the top and bottom equations together. $ 2y = 20 $ $ y = \dfrac{20}{2}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {7x+2.5y = 53}$ to find $x$ ${7x + 2.5}{(10)}{= 53}$ $7x+25 = 53$ $7x = 28$ $x = \dfrac{28}{7}$ ${x = 4}$ You can also plug ${y = 10}$ into $ {21x+9.5y = 179}$ and get the same answer for $x$ ${21x + 9.5}{(10)}{= 179}$ ${x = 4}$ There were $4$ teachers and $10$ students on the field trips.